Now we use that the morphism \(\psi \)  from Lemma REF  induces a natural weak equivalence \(\psi \colon |{-}| \circ S_e \overset{\sim }{\longrightarrow }\mathrm {D}\)  of functors \({{C}\mathrm {art}}\rightarrow {\Delta {T}\mathrm {op}}\)  (cf. Lemma REF ).
Since each of the functors \(F \colon {{C}\mathrm {art}}^\mathrm {op}\rightarrow {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  and \(\mathrm {D}, |{-}| \circ S_e \colon {{C}\mathrm {art}}\rightarrow {\Delta {T}\mathrm {op}}\)  are objectwise cofibrant, [1]} implies that \(\psi \)  induces a weak equivalence
\(B^{\Delta {T}\mathrm {op}}(-, {{C}\mathrm {art}}, \psi ) \colon B^{\Delta {T}\mathrm {op}}(-, {{C}\mathrm {art}}, |{-}| \circ S_e) \overset{\sim }{\longrightarrow }B^{\Delta {T}\mathrm {op}}(-, {{C}\mathrm {art}}, \mathrm {D}) = Re \circ Q^p\) 
